Problem #0194 Mechanics Sub-menu Problem #0196 Mechanics Main Menu Home Page

195.

Problem 3.43 (R)

Consider electromagnetic radiation to consist of photons, that is, particles of zero rest mass and of energy We have to show that the Doppler shift can be obtained from the transformation laws for the components of momentum and the energy.

Solution:             Click For PDF Version

Let us consider two inertial frames S and .Let be moving with respect to S in the direction of x-axis to the right with speed V. Let a source of light stationary in S emit photons of frequency . An observer in measure frequency of photons and find it to . With respect to observer in the source is receding away from S with speed V.

Energy of a photon of frequency is and its momentum .

Energy of this photon as measured by will be and its momentum .

The Lorentz transformation of energy and momentum is

Therefore, we have

And when the source and the observer are moving toward each other with speed V, we have

The relations between the emitted frequency and the observed frequency when the source and the observer have constant relative velocity derived above are the relativistic Doppler effect.